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Neutrino Spectrum Distortion Due to Oscillations and its BBN Effect Daniela Kirilova† 4 0 †Institute of Astronomy, Sofia 0 and Physique Theorique, ULB, Bruxelles 2 n a J Abstract 2 2 We study the distortion of electron neutrino energy spectrum due to oscillations 3 v with the sterile neutrino νe νs, for different initial populations of the sterile state ↔ 4 δN at the onset of oscillations. The influence of this spectrum distortion on Big 0 s 1 Bang Nucleosynthesis is analyzed. Only the case of an initially empty sterile state 9 0 was studied in previous publications. 2 TheprimordialabundanceofHe-4iscalculatedforallpossibleδN : 0 δN 1 0 s ≤ s ≤ / in the model of oscillations, effective after electron neutrino decoupling, for which h p the spectrum distortion effects on the neutron–proton transitions are the strongest. - p It is found that the spectrum distortion effect may be the dominant one not e h only in the case of small δNs, but also in the case of big initial population of νs. : v For example, in the resonant case it may play a considerable role even for very big i X δNs 0.8. ∼ r a keywords: cosmology, primordial nucleosynthesis, neutrino oscillations 1 Introduction Sterile neutrinos ν may be present at the onset of the nucleosynthesis epoch. There s may be different reasons for their production — they are naturally produced in GUT models [1], in models with large extra dimensions [2] and Manyfold Universe models [3], in mirror matter models [4]. They may be produced also in ν ν oscillations in the µ,τ s ↔ preceeding epoch 1 according to the most popular 4-neutrino mixing schemes [5]. The degree of population ofν , δN , may be different depending onthe concrete model s s of ν production. Hence, we will consider further on δN as a parameter. s s In recent years, strong constraints on the sterile neutrino impact in oscillations, ex- plaining atmospheric and solar neutrino anomalies, were obtained from the analysis of experimental oscillations data [6, 7, 8]. And although a pure sterile channel solution is excluded for any of the neutrino anomalies, the analysis of neutrino oscillation experimental data allows a certain, small fraction of sterile neutrino to participate into these oscillations. In the analysis [9] and [10] it was suggested that a small fraction of ν is not s only allowed but even desirable for solar neutrino data. Recent measurements of the cross section of 7Be(p,γ)8B reaction lead to a predicted total 8B neutrino flux by 13% larger than measured by SNO, which also may be providing evidence for sterile neutrino [11]. Therealsoexiststringentcosmologicalconstraintsonν producedinoscillations, based s on oscillations influence on BBN nucleosynthesis of 4He. For the case ν ν see µ,τ s ↔ for example refs. [12, 13] and for the ν ν see refs. [14, 13]. The constraints on e s ↔ electron–sterile neutrino oscillations [15, 16], excluded the active–sterile LOW solution to the solar neutrino puzzle, in addition to the already excluded in pioneer works (see for example ref.[17]) sterile LMA solution and sterile solution to the atmospheric neutrino anomaly. (For more detail about constraints on neutrino oscillations from cosmology, see refs. [18, 12].) Most of the cosmological constraints on active-sterile mixing were obtained in simple two-neutrino mixing schemes 2, for the case when the sterile neutrino state was initially 1For example, atmospheric and LSND neutrino data require oscillations with maximal mixings and massdifferencesδm2 10−3 eV2 andδm2 10−1 eV2,whichareeffectivebeforetheBBNepoch. atm ∼ LSND ∼ So, in many schemes owing to ν ν or ν ν oscillations, ν state may be partially thermalized µ s τ s s ↔ ↔ before the nucleosynthesis epoch. 2 Fordiscussionandcalculationofcosmologicalconstraintsinaspecific 4-neutrinomixing schemessee ref. [19, 20, 13] 2 empty at the epoch before oscillations became effective in the Universe evolution, δN = s N 3 = 0. N is the number of neutrino species in equilibrium. Since the presence ν ν − of a non-empty sterile state before oscillations was not considered in previous analysis of oscillations effects on the neutrino spectrum distortion and on BBN, in this work we address this question. We omit the assumption δN = 0 and explore the general case s δN = 0. s 6 In the ν ν oscillation case, δN = 0 present initially just leads to an earlier µ,τ s s ↔ 6 increase of the total energy density of the Universe, and it is straightforward to re-scale the existing constraints. In the ν ν oscillations case, however, the presence of ν at e s s ↔ the onset of oscillations influences also the kinetic effects of ν ν on BBN. Therefore, e s ↔ we chose ν ν oscillations case for exploring electron neutrino distortion, caused by e s ↔ oscillations, and its influence on nucleons freezing and on primordial 4He, Y for different p δN values in the range 0 δN 1. s s ≤ ≤ The spectrum distortion of the electron neutrinos We analyze the case of oscillations effective after neutrino decoupling, therefore further on we denote by δN the degree of population of the sterile neutrino state at active neutrino s decoupling (T 2 MeV). Here we explore ν spectrum distortion , considering the degree e ∼ of population of the sterile neutrino state δN as a free parameter and, varying its value s in the range [0,1] with a step 0.1. In the next section we calculate the kinetic effect of oscillations on primordial abundance of 4He for different δN . s The mixing of electron neutrino with ν has the following two types of effects on BBN: s (a) it leads to an increase of the energy density of the Universe [21], and (b) it changes the nucleons kinetics essential for n/p-freezing, through the depletion of electron neutrino number density, distortion of the equilibrium spectrum of ν , and pro- e duction of asymmetry between neutrinos and anti-neutrinos, all due to oscillations [18]. We will parametrize these kinetic effects and denote them δN further on. kin (a) The first effect is usually described by an increase of the effective number of the energy density degrees of freedom g = (30/π2)(ρ/T4). At the BBN epoch g = eff eff 10.75+7/4δN (T /T)4. Hence, this effect leads to a faster expansion rate H g1/2 and s ν ∼ eff 1/6 higher freezing temperature for nucleons T g , when nucleons were more abundant: f ∼ eff n/p exp[ ∆m/T ] f ∼ − This reflects into an overproduction of 4He, since it strongly depends on the n/p- freezing ratio: Y 2exp( ∆m/T )/[1+exp( ∆m/T )], where ∆m = m m 1.3 p f f n p ∼ − − − ∼ 3 MeV is the neutron–proton mass difference. This effect of g increase on BBN is well eff known [22]. The approximate fit to the exact calculations is: δY 0.013δN . The p s ∼ maximum helium overproduction corresponding to δN = 1 is 5%. s ∼ (b) The influence of the kinetic effects of oscillations on BBN is quite obvious, having in mind that: (i) oscillations take place between equilibrium electron neutrino and less populated sterile neutrino ensemble, that (ii) the oscillations probability is inversely proportional to the energy of neutrinos P δm2/E, so that neutrinos with different mo- ∼ menta start oscillating at different cosmic times, and that (iii) the proton density is bigger than the neutron one. Due to that, the neutrino energy spectrum n (E) may strongly ν deviate from its equilibrium form [23, 18]. In case oscillations proceed after the decoupling of active neutrinos, a strong spectrum distortion for both the electron neutrino and the anti-neutrino is possible. This spectrum distortion affects the kinetics of nucleons freezing - it leads to an earlier n/p-freezing and an overproduction of 4He yield. The effect can be easily understood having in mind that the distortion leads both to a depletion of the active neutrino number densities in favor of the sterile ones N : ν 2 N dEE n (E) ν ∼ Z ν and to a decrease of the mean neutrino energy. 3 This lowers the weak rates, governing nucleons transitions during neutrons freezing, with respect to their values in the standard BBN model, Γ N E¯2, and hence reflects into earlier freezing when neutrons were weak ∼ νe ν more abundant. So, helium is over-produced. The generation of neutrino-antineutrino asymmetry in the resonant oscillations has a subdominant effect on 4He. It slightly suppresses oscillations at small mass differences, leading to a decrease of helium overproduction. δN depend strongly on the initial population of the sterile neutrino at BBN. Larger kin δN decreases the kinetic effects, because the element of initial non-equilibrium between s the active and the sterile states is less expressed. Hence, for any specific value of δN it s is necessary to provide a separate analysis. In the case δN = 1 ν are in equilibrium (the sterile state is as abundant as the s s electron one), and hence the n–p kinetics does not feel the oscillations, δN = 0. The kin final effect is only due to the energy increase, i.e. δN = 1. tot 3Thedecreaseoftheelectronneutrinoenergyduetooscillationsintolowtemperaturesterileneutrinos, hasalsoanadditionaleffect: Duetothethresholdofthereactionconvertingprotonsintoneutrons,when neutrinos have lower energy than the threshold one, protonsare preferably produced, which may lead to 4 an under-production of He [24]. However, this turns to be a minor effect in the discussed oscillations model. 4 In the case δN = 0 the kinetic effect of oscillations was studied numerically for both s the resonant [15] and non-resonant [16] oscillation cases. In this case δN for given kin fixed mixing parameters reach their highest value, δNmax, as far as the non-equilibrium kin element — the difference between the sterile and active neutrino number densities at the beginning of oscillations — is the greatest. The overproduction of 4He may be enor- mous: up to 13.2% in the non-resonant oscillation case and up to 31.8% in the resonant one [25]. This corresponds effectively to a little more than 6 additional neutrino states δNmax 6. For the case δN = 0 the kinetic effects were carefully studied and strin- kin ∼ s gent cosmological constraints on the oscillation parameters were obtained on the basis of helium-overproduction, caused by the kinetic effects of oscillations [14]. Inthiswork,accountingforalltheeffects(a)and(b),wecalculateY (δN ,δm2,sin22θ) p s for 0 < δN 1 values, and reveal the dependence of δN on δN . s kin s ≤ We have analyzed the self-consistent evolution of the oscillating neutrino and the nucleons from the neutrino decoupling epoch at 2 MeV till the freezing of nucleons. ∼ We have followed the line of work described in detail in ref. [15], omitting the assumption for negligible density of the sterile neutrinos at the onset of ν ν oscillations. e s ↔ It is hardly possible to describe analytically, without some radical approximations, the non-equilibrium picture of active–sterile neutrino oscillations, producing non-equilibrium neutrino number densities and distorting the neutrino spectrum. Satisfactory precise analytical description was found only for the case of relatively fast oscillations proced- ing before neutrino freezing, with δm2 > 10−6 eV2 and small mixing angles [26, 13]. Therefore, we have provided a self-consistent numerical analysis of the evolution of the nucleons number densities n and the ones of the oscillating neutrinos ρ and ρ¯ in the n high-temperature Universe, using the following coupled integro-differential equations, the first equation describing the kinetics of the neutrino ensembles in terms of the density matrix of neutrino ρ and anti-neutrino ρ¯, the second equation – the kinetic evolution of the neutrons. ∂ρ(t) ∂ρ(t) = Hp + ν ∂t ∂p ν +i[ ,ρ(t)]+i√2G Q/M2 N [α,ρ(t)]+O G2 , (1) Ho F (cid:16)±L− W(cid:17) γ (cid:16) F(cid:17) (∂n /∂t) = Hp (∂n /∂p )+ n n n n +Z dΩ(e−,p,ν)|A(e−p → νn)|2[ne−np(1−ρLL)−nnρLL(1−ne−)] + + 2 dΩ(e ,p,ν˜) (e n pν˜) [n n (1 ρ¯ ) n ρ¯ (1 n )]. (2) −Z |A → | e+ n − LL − p LL − e+ 5 where α = U∗U , p is the momentum of electron neutrino, n stands for the number ij ie je ν density of the interacting particles, dΩ(i,j,k) is a phase-space factor, and is the ampli- A tude of the corresponding process. The plus sign in front of corresponds to the neutrino L ensemble, the minus sign - to the anti-neutrino ensemble. Mixing just in the electron sector is assumed: ν = U ν (l = e,s). The initial i il l condition for the neutrino ensembles in the interaction basis is of the form: 1 0 ρ = neq , ν 0 S   where neq = exp( E /T)/(1+exp( E /T)), while S measures the degree of population ν − ν − ν of the sterile state. is the free neutrino Hamiltonian. The ‘non-local’ term Q arises as a W/Z propa- o H gator effect, Q E T. is proportional to the fermion asymmetry of the plasma and ν ∼ L is essentially expressed through the neutrino asymmetries 2L +L + L , where L ∼ νe νµ ντ L (N N )/N and L d3p(ρ ρ¯ )/N . µ,τ ∼ µ,τ − µ¯,τ¯ γ νe ∼ LL − LL γ R The equations are for the neutrino and neutron number densities in momentum space, which allows to describe precisely the kinetic effects: spectrum distortion and neutrino- antineutrino asymmetry growth due to oscillations. For the description of the spectrum 1000 bins were used in the nonresonant oscillations case, and at least 5000 in the resonant one. These equations providea simultaneous account ofthedifferent competing processes, namely: neutrino oscillations, Universe expansion, neutrino forward scattering, nucleons transformations. The analysis was performed for all mixing angles ϑ and mass differences δm2 10−7 ≤ eV2. The analyzed temperature interval was [2.0,0.3] MeV, because at temperatures higher than 2 MeV the deviations from the standard BBN model without oscillations are negligible in the discussed model of oscillations. As expected, the spectrum distortion is less expressed when increasing the degree of population of the sterile neutrino state δN (Fig.1.). Correspondingly, the kinetic effect s on primordial nucleosynthesis should decrease. The results of our numerical analysis on spectrumdistortionatdifferentδN areillustratedintheFigs. 1a-c,wherethedependence s of the energy spectrum distortion of the electron neutrino on the initial population of the sterile state is shown. At each temperature we have plotted the spectrum for three different levels of initial population of the sterile neutrino, namely δN = 0.0,0.5,0.8. s 6 Figure 1a: The figure illustrates the degree of distortion of the electron neutrino energy spectrum x2ρ (x), where x = E/T at a characteristic temperature 1 MeV, caused by resonant LL oscillations with a mass difference δm2 = 10−7 eV2 and sin22ϑ = 0.1 for different initial sterile neutrino populations, correspondingly δN = 0 (the lower curve), δN =0.5 and δN = 0.8 (the s s s upper curve). The dashed curve gives the equilibrium neutrino spectrum for comparison. 7 Figures 1b,c: Distortion of the electron neutrino energy spectrum at a temperature 0.7 MeV (Fig.1b) and 0.5 MeV (Fig.1c) for the same parameters as for Fig.1a. 8 The oscillations parameters are δm2 = 10−7 eV2 and sin22ϑ = 0.1. For illustrating | | the evolution of the spectrum distortion we have presented it at characteristic temperatures 1 (Fig.1a), 0.7 (Fig.1b) and 0.5 MeV (Fig.1c). At each δN the characteristic be- s havior of the spectrum distortion due to oscillations is observed. Namely, since oscillation rate is energy dependent Γ δm2/E the low energy part of the spectrum is distorted first ∼ (as far as low energy neutrinos start to oscillate first) and later the distortion penetrates noticeably into the more energetic part of the spectrum. We have found that the neutrino energy spectrum n (E) may strongly deviate fromits ν equilibrium formduring alltheperiodofinterest (2 MeV– 0.3MeV) even forconsiderably large δN , and hence the spectrum distortion may constitute the dominant effect on the s overproduction of 4He. The kinetic effect For different δN we calculate precisely the n/p-freezing, essential for the production of s helium, down to temperature 0.3 MeV. Then we calculate Y , accounting adiabatically p for the following decay of neutrons till the start of nuclear reactions, at about 0.1 MeV. We have found that neutrino spectrum distortion effect on BBN is very strong even when there is a considerable population of the sterile neutrino state before the beginning of the electron–sterile oscillations. It always gives positive δN , which for a large range kin of initial sterile population values, are bigger than 1. The kinetic effects are the strongest for δN = 0: Ymax(δN ,δm2,sin22ϑ) = Y (0,δm2,sin22ϑ). They disappear for δN = 1, s p s p s when ν and ν states are in equilibrium, and the total effect reduces to the SBBN with e s an additional neutrino. In Fig.2 we present the frozen neutron number density relative to nucleons Xf = n Nf/N as a function ofthe sterile neutrino content at neutrino decoupling for a resonant n nuc and a nonresonant oscillation case. The oscillation parameters are δm2 = 10−7 eV2 and δm2 = 10−7 eV2 and sin22θ = 10−1. As far as δY /Y = δXf/Xf, the results are − p p n n representative for the overproduction of primordially produced helium. 9 Figure 2: The solid curves present frozen neutron number density relative to nucleons Xf = n Nf/N as a function of the sterile neutrino initial population. The dashed curves present only n nuc the kinetic effect, while the dotted curve presents the effect due to the energy density increase. The upper two curves (dashed and solid) correspond to the resonant case, the lower dashed and solid curves - to the nonresonant one. The dotted curve presents only the effect (a), due to the energy density increase Xf = f(δN ), the dashed curves present the pure kinetic effects (b) Xf = f(δN ), while n s n kin the solid lines give the total effect. The upper dashed and solid curves correspond to the resonant case, the lower ones to the non-resonant one. The analysis for these concrete oscillation parameters, shows that the overproduction ofhelium isstronglysuppressed withtheincrease ofδN fortheresonant case, while inthe s non-resonant case it increases with δN . This is a result of the fact that, in the resonant s case, the kinetic effects (b) due to the spectrum distortion are the dominant contribution to the overproduction of helium, even for very large degree of population of the sterile state, while in the non-resonant case the main contribution comes from the increase of degrees of freedom already at very small δN . An empirical approximation formula is: s δY = 0.013[δNmax(1 δN )+δN ], p kin − s s where δNmax is the value calculated in the case of oscillations with an initially empty kin sterile state, i.e. δN = δNmax(1 δN ) + δN . It is a good approximation for the tot kin − s s 10